Honestly, if we were even going to begin to entertain any notions even slightly related to this, I'd want to suggest that if we insist on having the death penalty as a punishment, then those who are put to death should, if suitable, become automatic organ donors.

Nah that's totally ridiculous. Death is as serious a punishment as possible and can never be appealed once delivered, there should never be any incentives to deliver the death sentence besides seeing that justice is done.

Let me be clear that I'm not advocating for capital punishment at all, I'm staunchly against it. But honestly, if we're going to decide that we're allowed to kill people in the name of justice, it's arguable that the deceased's bodies should be used to help save lives.

Well, lots of things are arguable to some extent, but really the idea that the state should materially benefit from murdering its own citizens is pretty difficult to defend in my opinion. The moment a judge starts to think "well, what about all the lives this prisoner's healthy organs will save?" when deciding whether or not to give a death sentence, your society is 100% fucked. If anything your hypothetical society should deliberately not take the organs of any one sentenced to death.

Absolutely not. If anything their bodies should be protected from any kind of harvesting as to not incentivize the state from handing out death penalties.

I believe the death penalty should be abolished anyway. But, disregarding the complexity of the implementation of such a policy, if the government is in the business of ending people's lives, I'd rather it be done to directly help people, instead of in a punitive fashion. If you think about it, the death penalty isn't really a punishment - some people commit heinous crimes specifically so they can be killed.

"Suicide by Justice" doesn't seem like it would be or is a popular enough form of suicide to warrant what you're suggesting. Your logic doesn't make much sense.